A Sophie Germain prime is a prime \(p\) for which \(2p + 1\) is also prime. When it is, the companion \(2p+1\) is called a safe prime.
They are named for Sophie Germain (1776–1831), who used them in the first general progress on Fermat's Last Theorem — proving Case 1 for all such primes — at a time when, as a woman, she had to correspond with Gauss under the male pseudonym M. LeBlanc.
Sophie Germain primes matter practically too: safe primes make the discrete logarithm problem hard, so cryptographic protocols like Diffie–Hellman favor moduli built from them. Long chains of repeated \(p \to 2p+1\) steps form Cunningham chains. It is conjectured but unproven that infinitely many Sophie Germain primes exist.